Black hole information via ADM reduction

Inyong Park

Philander Smith College

KIAS, Seoul Dec 2013

Contents 2/27 • 1. Black hole information (BHI) Review of BHI problem , black hole complementarity (BHC),

Firewall

• 2. Assessment Limitations of semiclassical description, quantum gravity effect: bleaching and/or blackening (potential presence of these

mechanisms will lead to a very different solution of BHI problem)

• 3. Reduction of BTZ to hypersurfaces of foliation

appearance of Liouville type theories

• 4. Conclusion

1 BHI

• Hawking discovered that a BH radiates (Hawking radiation) once quantum effects are taken into account

• Hawking radiation is thermal (this is modified in BHC)• Evolution of pure state of initial matter into thermal state of

radiation ; non-unitary evolution, but how? (BHI problem)• Different causes with the same effects. Hawking originally

suggested modification of quantum principles. (Later he conceded.)

• There are (at least) three stances one can take:1. Agree with Hawking’s original opinion: modification of

quantum principles2. Try to come up with an approach that accommodates BH evaporation and QM: BHC3. Keep QM but end up abandoning something else (Firewall contradicts with equivalence principle)

1 BHI

• Schwarzschild coordinate (stationary observer) vs Kruskal coordinate (free-falling observer)• Scalar field theory in these two coordinates, creation and

annihilation operators.

• Two different Fock vacuua :Schwarzschild vacuum |S> vs Kruskal vacuum |K>

• They are inequivalent In particular, the Kruskal vacuum appears as a radiating BH to a

Schwarzschild observer (stationary observer) (One can see this by looking at expectation value of the stress-energy

tensor; it displays blackbody radiation at a certain temperature)(This is a long-known fact, known much before complementarity.)• BH radiates and eventually evaporates with information information loss ?

According to Firewall argument

• Even for a free-falling observer, |K> cannot be a vacuum state if the Hawking radiation is taken to be pure as one assumes in BHC.

1 BHIblack hole complementary• L. Susskind, L. Thorlacius and J. Uglum; C.

R. Stephens, G. 't Hooft and B. F. Whiting• Unitarity is one of the foundational principles of quantum physics• Assume: somehow the BH evaporation process is unitary (What is given up is locality)

a set of 4 postulates• 1. Unitary evolution: in particular, there exists a unitary S-matrix

which describes the evolution from infalling matter to outgoing Hawking-like radiation. (Purity of Hawking radiation)

• 2. Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations. (Semiclassical description of geometry)

• 3. To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass M is the exponential of the Bekenstein entropy.

• 4. A freely falling observer experiences nothing out of the ordinary when crossing the horizon.

(No drama (no trouble), equivalence principle)

1 BHI

Firewall

• Ahmed Almheiri, Donald Marolf, Joseph Polchinski, James Sully

• Infalling observer experiences violent horizon due to outgoing high energy radiation,

• Firewall: an infalling observer will burn up at the horizon• Violation of Equivalent Principle: Infalling observer must see smooth event horizon according to

Equivalence Principle

• Many and ongoing debates after their paper

1 BHIFirewallIn a few words: Firewall is Violent horizon due to high energy radiation,

inconsistent with equivalence principle

(high energy radiation)

1 BHI

Inconsistent with Equivalence Principle according to which

Smooth horizon

1 BHI

Firewall• 3 out of 4 BHC postulates cannot all be true: • Purity of Hawking radiation, Emission of information from

horizon (semiclassical description), No drama cannot all be true• At least for sufficiently old black holes (after the Page time: the

black hole has emitted half of its initial Bekenstein-Hawking entropy), there must be a firewall (a violent horizon)

• The relevance of Page time is technical; we will not be concerned

• (earlier related observation by S. L. Braunstein: energetic curtain)

1 BHI• Divide the BH system into 3 parts:Mode : early (before Page time) Hawking radiationMode : late (after Page time) Hawking radiation Mode : the interior “partners” of the late Hawking radiation

Mode expansion

Modes in Schwarzschild coordinates: Modes in Kruskal coordinates:

(These are schematic notations)

1 BHI

Purity of Hawking radiation implies

Consider an outgoing Hawking mode in the late radiation.

Stationary observer measures on the early radiation. This projectsthe state of late radiation into an eigenstate of

Next consider an infalling observer and the associated set of infalling modes . The afore-mentioned eigenstate of cannot be a vacuum of the number operator of a mode . This is because the Schwarzschild vacuum and Kruskal vacuum are inequivalent.

Therefore the infalling observer sees radiation and this radiation has high frequency due to blueshift (to see blueshift, propagates backwards in time)

1 BHI From slightly different angle:

• The purity of the Hawking radiation implies

; the late radiation is fully entangled with the early radiation

No drama for the infalling observer implies late radiation is fully entangled with the BH, i.e., the modes behind the event horizon

• Because entanglement can not be shared (otherwise information cloning), the late radiation can not be entangled with the black hole.

• the latter property (,i.e., entanglement between late radiation and BH) is believed to be required for the BH quantum state to locally look like Minkowski vacuum near the horizon, one concludes that the infalling observer cannot pass through the horizon smoothly, i.e., without experiencing a drama.

• The dramatic event is high energy (due to blue-shift) radiation

2 Assessment 14/27

Let us pause and assess the situation• Unrenormalizability of 4D gravity semiclassical description• Limitations of semiclassical description Semiclassical description; geometry is fixed, i.e.,

nonfluctuating, usually free (,i.e., quadratic) QFT description, gravity theories are interacting theories, gravitational Bremsstrahlung might be relevant in various circumstances but semiclassical description is not adequate for quantum field theory interactions

2 Assessment• In semiclassical picture, information enters BH intact and BH

remains black throughout• In full quantum gravity description: Information bleaching and

blackening through meta-black states• Presence of these phenomena would have direct implications for

BHI itself but may also shed light on Firewall • Study of BHI in a different approach (quantum gravity of

hypesurface of foliation)

2 Assessment• Quantization issue can be “bypassed” by reduction to lower

dimensions, reduction to 3D

2 Assessment• Interestingly: informtion obliteration observed by Susskind, Thorlacius

and Uglum (hep-th/9306069)

“~” means related by S-matrix

therefore

Must be independent of the initial state ! “obliteration of information”

This is in the semiclassical picture; in the full quantum picture it should be information bleaching

3 Reduction to hypersurface of foliation

• Variation of Kaluza-Klein reduction but not the same

3 Reduction to hypersurface of foliation 19/27

• AdS/CFT is bulk/bd correspondence with bd at r=infinity• Bulk/bd correspondence seems to be quite general phenomenon

by now• Could bulk/bd surface correspondence be generalized to

“bulk/hypersurface correspondence”?• The answer seems affirmative. The former may be derived from

the latter • Should examine physics on the hypersurface• The concept of foliation is important • Will apply the reduction scheme to BTZ BH spacetime• Employ ADM decomposition: frequently used initial value problem and numerical relativity, Hamiltonian formulation, time evolution

3 Reduction of BTZ to hypersurfaces of foliation

r

r = r0

Bulk as a collection of hypersurfaces (foliation),motivated by derivation of AdS/CFT

hypersurfaces of foliation (ex: onion)

Is this approach useful?ADM reduction applied to IIB setup M. Sato and A. Tsuchiya (hep-th/0211074) E. Hatefi, A. J.Nurmagambetov and IYP (1210.3825)

Consider 10D IIB supergravity and compactify IIB supergravity on S^5Þ 5D AdS gravityÞ Solve its field equation after going through ADM splitting and

Hamilton-Jacobi formulationÞ The field equation takes a form of partial differential eq of the

principal functionÞ The solution of the PDE can be written as a 4D action of gauge field

in the curved backgroundÞ Seems to be a direct verification of “abelian AdS/CFT”

(a) (b)

3 Reduction of BTZ to hypersurfaces of foliation

3 Reduction of BTZ to hypersurfaces of foliation

• Consider

• BTZ black hole solution (non-rotating)

• ADM formulation : 1 (r direction) + 2 (i directions)

• The original action can be re-expressed as

• n_r: lapse function, N_i: shift function

3 Reduction of BTZ to hypersurfaces of foliation

• Set But be careful: this does not mean is a scalar.Eventually we go to 2D; there it should be ok; reduce according

to

Rescale

Naively, the system seems to reduced to

• Not so fast, what about the virtual boundary effects?• Does the system satisfy the “reduced BH solution”?

rbulk

boundary

Region I

Region II

Virtual boundary

3 Reduction of BTZ to hypersurfaces of foliation

• With the bd effects

• Gauge-fix the metric (which induces the Virosora constraint)

• The final form of the action is

4 Conclusion

• Have reviewed BHI problem and BHC• Have introduced Firewall argument• It may well be the inadequacy of semiclassical description that is

causing problems: geometry is fixed, i.e., non-fluctuating, usually free (,i.e., quadratic) QFT description

• Studying information bleaching mechanism bleaching should be interesting

• ADM reduction of BTZ spacetime => Interacting 2D QFT, Liouville type theory

• One can use Liouville theory to study various aspects of BTZ BH